Theoretically, we can show that, if \(\mathbb {E}[I - C_{t,j}]\)  is invertible, then \(\Vert \mathbb {E}[C_{t,j}]\Vert <1\) .
This implies that \(\Vert \mathbb {E}[\tilde{C}_t\mid {x}_t, {\lambda }_t]\Vert \)  vanishes to zero exponentially in \(\tau \) , and further implies that \(\Vert \mathbb {E}[{z}_{t,\tau } - {\tilde{z}}_t\mid {x}_t, {\lambda }_t, \xi _t]\Vert \)  vanishes to zero exponentially in \(\tau \)  due to (REF ).
Moreover, if \(\mathbb {E}[I - C_{t,j}]\)  is positive definite, a similar guarantee holds for \(\mathbb {E}[\Vert {z}_{t,\tau } - {\tilde{z}}_t\Vert \mid {x}_t, {\lambda }_t, \xi _t]\) .
We refer to [1]} for more details, and we state necessary results later (cf. Lemma REF ) to make our paper self-contained.
